Mathematica Bohemica, first online, pp. 1-18


A power of a meromorphic function sharing two small functions with a derivative of the power

Lahiri Indrajit, Sujoy Majumder

Received July 8, 2020.   Published online November 22, 2021.

Abstract:  In connection to a conjecture of W. Lü, Q. Li and C. Yang (2014), we prove a result on small function sharing by a power of a meromorphic function with few poles with a derivative of the power. Our results improve a number of known results.
Keywords:  meromorphic function; derivative; small function
Classification MSC:  30D35
DOI:  10.21136/MB.2021.0121-20

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References:
[1] R. Brück: On entire functions which share one value CM with their first derivative. Result. Math. 30 (1996), 21-24. DOI 10.1007/BF03322176 | MR 1402421 | Zbl 0861.30032
[2] T. Cao: On the Brück conjecture. Bull. Aust. Math. Soc. 93 (2016), 248-259. DOI 10.1017/S000497271500115X | MR 3480935 | Zbl 1379.30016
[3] Z.-X. Chen, K. H. Shon: On conjecture of R. Brück concerning the entire function sharing one value CM with its derivative. Taiwanese J. Math. 8 (2004), 235-244. DOI 10.11650/twjm/1500407625 | MR 2061691 | Zbl 1062.30032
[4] J. Clunie: On integral and meromorphic functions. J. Lond. Math. Soc. 37 (1962), 17-27. DOI 10.1112/jlms/s1-37.1.17 | MR 0143906 | Zbl 0104.29504
[5] G. Frank: Eine Vermutung Von Hayman über Nullslellen meromorpher Funktionen. Math. Z. 149 (1976), 29-36. (In German.) DOI 10.1007/BF01301627 | MR 0422615 | Zbl 0312.30032
[6] G. G. Gundersen, L.-Z. Yang: Entire functions that share one value with one or two of their derivatives. J. Math. Anal. Appl. 223 (1998), 88-95. DOI 10.1006/jmaa.1998.5959 | MR 1627356 | Zbl 0911.30022
[7] W. K. Hayman: Meromorphic Functions. Oxford Mathematical Monographs. Clarendon Press, Oxford (1964). MR 0164038 | Zbl 0115.06203
[8] I. Lahiri: Weighted sharing and uniqueness of meromorphic functions. Nagoya Math. J. 161 (2001), 193-206. DOI 10.1017/S0027763000027215 | MR 1820218 | Zbl 0981.30023
[9] I. Lahiri: Weighted value sharing and uniqueness of meromorphic functions. Complex Variables, Theory Appl. 46 (2001), 241-253. DOI 10.1080/17476930108815411 | MR 1869738 | Zbl 1025.30027
[10] S. Lin, W. Lin: Uniqueness of meromorphic functions concerning weakly weighted-sharing. Kodai Math. J. 29 (2006), 269-280. DOI 10.2996/kmj/1151936441 | MR 2247436 | Zbl 1105.30017
[11] F. Lü, H. Yi: The Brück conjecture and entire functions sharing polynomials with their $k$-th derivatives. J. Korean Math. Soc. 48 (2011), 499-512. DOI 10.4134/JKMS.2011.48.3.499 | MR 2815888 | Zbl 1232.30024
[12] W. Lü, Q. Li, C. Yang: On the transcendental entire solutions of a class of differential equations. Bull. Korean Math. Soc. 51 (2014), 1281-1289. DOI 10.4134/BKMS.2014.51.5.1281 | MR 3267229 | Zbl 1300.30063
[13] S. Majumder: A result on a conjecture of W. Lü, Q. Li and C. Yang. Bull. Korean Math. Soc. 53 (2016), 411-421. DOI 10.4134/BKMS.2016.53.2.411 | MR 3483447 | Zbl 1343.30025
[14] C.-C. Yang, H.-X. Yi: Uniqueness Theory of Meromorphic Functions. Mathematics and its Applications (Dordrecht) 557. Kluwer Academic Publishers, Dordrecht (2003). DOI 10.1007/978-94-017-3626-8 | MR 2105668 | Zbl 1070.30011
[15] L.-Z. Yang, J.-L. Zhang: Non-existence of meromorphic solutions of Fermat type functional equation. Aequationes Math. 76 (2008), 140-150. DOI 10.1007/s00010-007-2913-7 | MR 2443466 | Zbl 1161.30022
[16] J. Zhang: Meromorphic functions sharing a small function with their derivatives. Kyungpook Math. J. 49 (2009), 143-154. DOI 10.5666/KMJ.2009.49.1.143 | MR 2527380 | Zbl 1177.30040
[17] J. Zhang, L. Yang: A power of a meromorphic function sharing a small function with its derivative. Ann. Acad. Sci. Fenn., Math. 34 (2009), 249-260. MR 2489025 | Zbl 1177.30041
[18] J.-L. Zhang, L.-Z. Yang: A power of an entire function sharing one value with its derivative. Comput. Math. Appl. 60 (2010), 2153-2160. DOI 10.1016/j.camwa.2010.08.001 | MR 2719737 | Zbl 1205.30033

Affiliations:   Indrajit Lahiri, Department of Mathematics, University of Kalyani, Kalyani, West Bengal 741235, India, e-mail: ilahiri@hotmail.com; Sujoy Majumder, Department of Mathematics, Raiganj University, Raiganj, West Bengal 733134, India, e-mail: sm05math@gmail.com, sujoy.katwa@gmail.com


 
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